That depends on what chat you want to save. There are chat logs saved such as PM/whispers if you have that option turned on in game. From what I can tell there are no logs for common/squad/faction chat.

If you're in a chatbox and click 'save chat history', it saves to a file...granted, that file is barely readable in notepad, but it is readable. Under the PWI folder, then userdata > Msg > one of the number folders. I tried to paste it in here to give an example, but too many unknown characters. ._.

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